High School Practice Problems: Strategies
HS-9a) Strategy: Guess, check and revise Answer: 533
There is a banana plantation located next to a desert. The plantation owner must transport the bananas to market by camel across a 1,000 mile stretch of desert. The owner has only one camel. It carries a maximum of 1000 bananas at any one time, and it eats one banana every mile it travels. The plantation produced 3,000 bananas. What is the largest number of bananas that can be delivered to market? Explain your answer in detail.
HS-8c) Strategy: Draw a picture Answer: same
A large water glass is filled with pure water. A small wine glass is filled with pure wine. A spoonful of water is taken from the water glass, dumped into the wine glass and stirred thoroughly. Then a spoonful of the mixture in the wine glass is removed and dumped back into the water glass. Each glass started with a pure liquid, water or wine, and now contains some impurity either water or wine. Is there now more water in the wine or more wine in the water? Explain your answer in detail.
HS-8b) Strategy: Draw a picture This is a hard problem. Use it only if you can devote several days to it. E-mail us if you get a nice answer.
Consider a geoboard. A square path is a closed path in the shape of a square with edges parallel to the edges of the geoboard. Find an equation that predicts the minimum number of pegs that must be painted red so that every square path on an n-peg geoboard has at least one red peg along an edge or at a corner. Explain your answer in detail.
HS-8a) Strategy: Draw a picture Answer: 9 sides can be done. We think 10 can’t. On a 2X2 board, you can get only a triangle or square, so size matters.
A convex polygon has the property that any line segment joining any two points of the figure does not contain any points that lie outside the figure. What is the largest number of sides possible for a convex polygon made on a 5-peg by 5-peg geoboard? Does the size of the geoboard influence the result? Explain your answer in detail.
HS-7b) Strategy: Act out using objects Answer: 1-6-2-10-3-7-4-9-5-8
Arrange cards 1-10 so that this happens: The top card is a 1 and should be placed face up on the table; put the next card on the bottom of the deck; place the 3rd card , which should be a 2, face up on the table; put the next to the bottom of the deck. Continue until all the cards are in order on the table top. Explain your answer in detail.
HS-7a) Strategy: Act out the problem or use objects (and collect some data) This is a hard problem. Use it only if you can devote several days to it. If there are n knights, you get the right seat number by subtracting off the largest power of two strictly less than n and then multiplying by two.
This is a fictionalized historical problem. King Arthur wanted to decide who was the fittest to marry his daughter, and chose this method. When all his knights were seated at the round table, he entered the room, pointed to one knight, and said: "You live." The knight seated next wasn't so fortunate. "You die," said King Arthur, chopping off his head. To the third knight he said: "You live," and to the fourth, he said: "You die," chopping off his head. He continued doing this around and around the circle, chopping off the head of every other living knight, until just one was left. This remaining knight got to marry the daughter, but, as legend goes, he was never quite the same again. Find a pattern so you can predict where to sit (to live) no matter how many people are seated in the circle. Explain your answer in detail.
HS-5b) Strategy: Make a table, chart, or organized list Answer: 1, 3, 9, 27
1 weight can only weigh 1 integer
2 weights can weigh 1 + 2(1) different integers = 31
3 weights can weigh 1 + 2(1+3) different integers = 9 = 32
4 weights can weigh 1 + 2(1+3+9) different integers = 27 = 33
5 weights can weigh 1 + 2(1+3+9+27) different integers = 81 = 34
6 weights can weigh 1 + 2(1+3+9+27+81) different integers = 243 = 35
… and so on.
A 40 kg rock was used to weigh things on a balance scale. The rock was dropped and it broke into four pieces. Using combinations of the four rocks all the whole number weights from 1 to 40 kg could be weighed on the balance scale. What are the four weights? Explain your answer in detail.
HS-5a) Strategy: Make a table, chart, or organized list Answer: 39
Using any number of darts on a target with values 11 and 5, what is the largest score you cannot get? Explain your answer in detail.
HS-4a) Strategy: Look for a pattern Answer: The perfect cubes in order. Partial proof idea: We’re adding n numbers that average to n squared.
Consider the following interesting pattern: 1 = 1; 3 + 5 = 8; 7 + 9 + 11 = 27; 13 + 15 + 17 + 19 = 64; 21 + 23 + 25 + 27 + 29 = 125. State the generalization suggested by these examples, express it in suitable mathematical notation, and then prove it. Explain your answer in detail.
HS-3b) Strategy: Use logical reasoning Answer: 9
Maria and Lisa are playing a game. At the end of each game, the loser gives the winner a penny. After a while, Maria has won 3 games, and Lisa has 3 more pennies than she did when she began. How many games did they play? Explain your answer in detail.
HS-3a) Strategy: Use logical reasoning Answer: Arthur Gaucher - group of stores - 3 miles - logged - shorts; Bertha Eggleston - shopping all - 5 miles - neighbor's care - power mower; Carlo Hinnel - local store - 1 block - family car - lumber; Donna Friar - downtown - 10 miles - bus - slacks
Arthur, Bertha, Carlo, and Donna, whose last names are Eggleston, Friar, Gaucher, and Hinnel, each went shopping at a different place (downtown, a local store, a shopping mall, a small group of stores). The distances of their homes from the stores are 1 block, 3 miles, 5 miles, and 10 miles. Their transportation means were a bus, the family car, jogging, and a neighbor's car. Each person purchased only one kind of item (lumber, power mower, shorts, slacks). Sort out the clues below and match up everything. 1) Eggleston's purchase was too heavy and clumsy for her to manage alone, so she went with a neighbor in the neighbor's car. 2) Gaucher, who did not go to the local store, wore his purchase home. 3) The shopping mall was closer than the store where the slacks were bought but farther than the store where the jogger went. 4) The man who bought the lumber did not go as far as the jogger. 5) Bertha went farther than Hinnel but only half as far as the woman who took the bus downtown. 6) Carlo did not buy the shorts. Explain your answer in detail.
HS-2a) Strategy: Work backward Answer: 44
On the way home from school, Sally likes to eat peanuts. One day, just as she was reaching into her sack, a hideous, laughing creature jumped into her path, identified itself as a pig eyes, and grabbed her sack. It stole half of her peanuts plus two more. A bit shaken, Sally continued home. Before she had a chance to eat even one peanut, another horrid creature jumped into her path, and also stole half of her peanuts plus two more. Upset, she continued on. (What else could she do?) But before she had a chance to eat even one peanut, another of these tricksters jumped out and did the very same thing -- took half her peanuts plus two more. Now there were only two peanuts left in Sally's sack. She was so despairing, she sat down and began to sob. The three little pig eyes reappeared, feeling some sense of remorse, and told her they would return all her peanuts to her if she told them how many she had altogether when she started. How many peanuts had been in Sally's sack? Explain your answer in detail.
Read mathematical symbols, diagrams, graphs and text; and interpret, organize, clarify, and refine mathematical information from both written and oral sources for a given purpose.
Use appropriate symbols, diagrams, graphs, and vocabulary to summarize and communicate mathematical ideas and reasoning with precision and efficiency in ways appropriate for the audience and purpose intended.
Use inductive reasoning to form conjectures, use deductive reasoning to prove a valid conjecture, and develop a counterexample to refute an invalid conjecture. The focus should be on knowing when it may be appropriate to test or demonstrate an assumption or conjecture using examples, and when to prove an assumption or conjecture for all cases.
Explain and demonstrate the importance of generalizations in mathematics and the role of generalizations in inductive and deductive reasoning.
Synthesize mathematical information from multiple sources to draw a conclusion or evaluate the conclusions of others, analyze a mathematical argument, and recognize flaws or gaps in reasoning.
Analyze a problem, synthesize the information, determine how to represent the problem mathematically, and identify any implicit and explicit assumptions that have been made. Representations may include drawings, figures, or concrete models as well as equations, tables, and graphs.
Solve problems using an appropriate problem-solving process, including determining relevant information, formulating one or more strategies, solving the problem, verifying the solution, evaluating the solution for reasonableness, and interpreting the meaning of the solution in the context of the original problem.
Generalize a solution strategy for a single problem to a class of related problems.
GUESS, CHECK, AND REVISE (HSS-9 …)
Guessing and checking is helpful when a problem presents large numbers or many pieces of data, or when the problem asks students to find one solution but not all possible solutions to a problem. When students use this strategy, they guess the answer, test to see if it is correct and if it is incorrect they make another guess using what they learned from the first guess. In this way, they gradually come closer and closer to a solution by making increasingly more reasonable guesses. Students can also use this strategy to get started, and may then find another strategy which can be used.
DRAW A PICTURE (HSS-8 …)
For some students, it may be helpful to use an available picture or make a picture or diagram when trying to solve a problem. The representation need not be well drawn. It is most important that they help students understand and manipulate the data in the problem.
ACT IT OUT OR USE OBJECTS (HSS-7 …)
Some students may find it helpful to act out a problem or to move objects around while they are trying to solve a problem. This allows them to develop visual images of both the data in the problem and the solution process. By taking an active role in finding the solution, students are more likely to remember the process they used and be able to use it again for solving similar problems.
MAKE AND USE AN ORGANIZED LIST, TABLE, CHART OR GRAPH (HSS-5 …)
Making an organized list, table, chart or graph helps students organize their thinking about a problem. Recording work in an organized manner makes it easy to review what has been done. Students keep track of data, spot missing data, and identify important steps that must yet be completed. It provides a systematic way of recording computations. Patterns often become obvious when data is organized. This strategy is often used in conjunction with other strategies.
LOOK FOR A PATTERN (HSS-4 …)
A pattern is a regular, systematic repetition. A pattern may be numerical, visual, or behavioral. By identifying the pattern, students can predict what will "come next" and what will happen again and again in the same way. Sometimes students can solve a problem by recognizing a pattern, but often they will have to extend a pattern to find a solution. Making a number table often reveals patterns, and for this reason is frequently used in conjunction with looking for patterns.
USE LOGICAL REASONING (HSS-3 …)
Logical reasoning is really used for all problem solving. However, there are types of problems that include or imply various conditional statements such as, "if.. then," or "if.. then.. else," or "if something is not true, then...” The data given in the problems can often be displayed in a chart or matrix. This kind of problem requires formal logical reasoning as a student works his or her way through the statements given in the problem.
WORK BACKWARD (HSS-2 …)
To solve certain problems, students must make a series of computations, starting with data presented at the end of the problem and ending with data presented at the beginning of the problem.
SOLVE A SIMPLER OR A SIMILAR PROBLEM (HSS-1 …)
Making a problem simpler may mean reducing large numbers to small numbers, or reducing the number of items given in a problem. The simpler representation of the problem may suggest what operation or process can be used to solve the more complex problem.